Question

Differentiate log (x e^x) w.r.t. x log x.

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Solution

Let space space straight y equals log left parenthesis xe to the power of straight x right parenthesis comma space straight u equals xlog space straight x Now space straight y equals log space straight x plus log space straight e to the power of straight x therefore space space space straight y equals log space straight x plus straight x therefore space space space dy over dx equals 1 over straight x plus 1 therefore space space space dy over dx equals fraction numerator 1 plus straight x over denominator straight x end fraction Also space space space space straight u equals straight x space log space straight x space space space space space space space du over dx equals straight x 1 over straight x plus log space straight x.1 therefore space space space du over dx equals 1 plus log space straight x Now space dy over du equals fraction numerator begin display style dy over dx end style over denominator begin display style du over dx end style end fraction equals fraction numerator 1 plus straight x over denominator straight x end fraction cross times fraction numerator 1 over denominator left parenthesis 1 plus log space straight x right parenthesis end fraction therefore space space space space space dy over du equals fraction numerator 1 plus straight x over denominator straight x left parenthesis 1 plus log space straight x right parenthesis end fraction

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